Abstract. We present an extensive survey of bijective proofs of classical partitions identities. While most bijections are known, they are often presented in a different, sometimes unrecognizable way. Various extensions and generalizations are added in the form of exercises.

We utilize Dyson’s concept of the adjoint of a partition to derive an infinite family of new polynomial analogues of Euler’s Pentagonal Number Theorem. We streamline Dyson’s bijection relating partitions with crank ≤ k and those with k in the Rank-Set of partitions. Also, we extend Dyson’s adjoint of a partition to MacMahon’s “modular ” partitions with modulus 2. This way we find a new combinatorial proof of Gauss’s famous identity. We give a direct combinatorial proof that for n> 1 the partitions of n with crank k are equinumerous with partitions of n with crank −k.

New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.

. For any fixed integer k 2, we define a statistic on partitions called the k- rank. The definition involves the decomposition into successive Durfee squares. Dyson's rank corresponds to the 2-rank. Generating function identities are given. The sign of the k-rank is reversed by an involution which we call k-conjugation. We prove the following partition theorem: the number of self-2k-conjugate partitions of n is equal to the number of partitions of n with no parts divisible by 2k and the parts congruent to k (mod 2k) are distinct. This generalizes the well-known result: the number of self-conjugate partitions of n is equal to the number of partitions into distinct odd parts. 1. Introduction Let p(n) denote the number of unrestricted partitions of n [A2]. Ramanujan discovered and later proved p(5n + 4) j 0 (mod 5); (1.1) p(7n + 5) j 0 (mod 7); (1.2) p(11n + 6) j 0 (mod 11): (1.3) Dyson [D1], [D3] discovered remarkable combinatorial interpretations of (1.1) and (1.2). He defined an...

Abstract. We discuss conjugation and Dyson’s rank for overpartitions from the perspective of the Frobenius representation. More specifically, we translate the classical definition of Dyson’s rank to the Frobenius representation of an overpartition and define a new kind of conjugation in terms of this representation. We then use q-series identities to study overpartitions that are self-conjugate with respect to this conjugation. 1.

Abstract. Let spt(n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt(n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain explicit Ramanujantype

Abstract. We study two types of crank moments and two types of rank moments for overpartitions. We show that the crank moments and their derivatives, along with certain linear combinations of the rank moments and their derivatives, can be written in terms of quasimodular forms. We then use this fact to prove exact relations involving the moments as well as congruence properties modulo 3, 5, and 7 for some combinatorial functions which may be expressed in terms of the second moments. Finally, we establish a congruence modulo 3 involving one such combinatorial function and the Hurwitz class number H(n). 1.

Abstract. We give a combinatorial proof of the first Rogers-Ramanujan identity by using two symmetries of a new generalization of Dyson’s rank. These symmetries are established by direct bijections.

In his own contribution to this volume, George Andrews has touched on a number of themes in his research by looking at the early influences on him of Bailey, Fine, MacMahon, Rademacher and Ramanujan. In this paper, I propose to present a survey of his work organized on a

In [10] various conjectural identities between the ranks and the cranks of partitions modulo 8, 9 and 12 were proposed. These conjectures have now all been proved [14, 15, 11], with three exceptions. Here, we prove these three exceptions. Methods similar to those used here may be used to establish all the theorems and conjectures of [10].